This invention relates to methods and apparatus for pricing financial derivatives.
Current pricing models for financial derivatives are based on two pricing methods: (1) the “Classic Arbitrage Argument” (CAA), or (2) “Monte-Carlo” method. The CAA states the return on a perfectly-hedged investment portfolio is precisely equal to the return on U.S. Treasury bonds. Under the CAA method, it is assumed that investors are capable of continuously hedging their portfolios in infinitesimal increments with no costs. Moreover, the CAA method assumes that the parameters of the model do not change during the lifetime of the portfolio and that the markets will always precisely follow the pricing model.
Pricing models based on the CAA method are often referred to as “Risk Neutral Pricing” (RNP) models, because no risks are assumed by the CAA method. Under the CAA method, management of all risks and all discrepancies between the pricing model and the real market is passed to the human users of the model. RNP models are implemented for production in a form of numerical solutions to partial differential equations via finite difference schemes, finite element methods, closed form formulas, or by various tree type methods, including binomial and trinomial ones. For example, the Black-Scholes pricing method is an example of a method that is based on the CAA method and idealistic RNP assumptions.
Monte Carlo pricing models, on the other hand, are based on statistical and probabilistic evaluation of possible future price scenarios. These scenarios are typically generated with the aid of random number generators. Evaluation of derivative prices is modeled via the CAA, discounted payoff method, or a combination of both. In either case, the value of the risk does not explicitly enter the evaluation formula.
Therefore, RNP pricing models do not include market risks and costs in the valuation of a financial derivative. As a result, these pricing models generate valuations of financial derivatives that are not completely consistent with actual market behavior. Although the risks have been reflected in the market prices by various non-mathematical methods, the lack of appropriate mathematical modeling has precluded the explicit inclusion of risks into these pricing and hedging models.